MEAN-VALUE PROPERTY AND SUBDIFFERENTIAL CRITERIA FOR LOWER SEMICONTINUOUS FUNCTIONS

We define an abstract notion of subdifferential operator and an associ ated notion of smoothness of a norm covering all the standard situatio ns. In particular, a norm is smooth for the Gateaux (Frechet, Hadamard , Lipschitz-smooth) subdifferential if it is Gateaux (Frechet, Hadamar d, Lipschitz) smooth in the classical sense, while on the other hand a ny norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the conn ection between the smoothness of norms and the subdifferentiability pr operties of functions. The proof relies on an adaptation of the ''smoo th'' variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, co ne-monotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for s pecific subdifferentials.